On motion by curvature of a network with a triple junction

Abstract

We numerically study the planar evolution by curvature flow of three parametrised curves that are connected by a triple junction in which conditions are imposed on the angles at which the curves meet. One of the key problems in analysing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularisation parameter shrinks to zero. Convergence of the new semi-discrete finite element scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularisation parameter on the properties of scheme and the accuracy of the results is numerically investigated.

Convergence of a scheme for an elastic flow with tangential mesh movement

“…1. Picking up an idea from [25] (which is on planar curves) we may split up the velocity in its normal and tangential part, x t = P x t + (x t • τ )τ and then scale all terms with tangential contributions with a small parameters 0 < ε 1. Equivalently, we consider a weighted L 2 gradient flow for…”

“…In effect, the tangential movement of mesh points that is beneficial for the mesh quality and due to the Dirichlet energy contribution in (1.4) is kept. We have not analyzed this idea but refer to [25], where a network of curves moving by curve shortening flow is investigated. Let us note that, there, constants in error estimates depend in an unfavorable way on ε, as does the conditioning of the linear systems in the fully discrete scheme.…”

Convergence of a scheme for elastic flow with tangential mesh movement

“…1. Picking up an idea from [22] (which is on planar curves) we may split up the velocity in its normal and tangential part, x t = P x t + (x t • τ )τ and then scale all terms with tangential contributions with a small parameters 0 < ε 1. Equivalently, we consider a weighted L 2 gradient flow for…”

“…In effect, the tangential movement of mesh points that is beneficial for the mesh quality and due to the Dirichlet energy contribution in (1.4) is kept. We have not analyzed this idea but refer to [22], where a network of curves moving by curve shortening flow is investigated. Let us note that, there, constants in error estimates depend in an unfavorable way on ε, as does the conditioning of the linear systems in the fully discrete scheme.…”

Learning phase field mean curvature flows with neural networks

“…The numerical approximation is however more difficult in dimensions higher than 2 for the method can hardly handle topological changes. The processing of singularities is difficult even in dimension 2, see for instance the recent work [6].…”